Here we present a brief review on real and complex representations of a finite group \(G\) . For details and proofs we refer the reader to [1]}.
A real (or complex) representation means the underlying vector space is real (or complex).
For a complex representation \((\pi ,V)\)  of \(G\)  we write \((\pi _{\mathbb {R}},V_{\mathbb {R}})\)  for the realization of \(\pi \) . This simply means that we forget the complex structure on \(V\)  and regard it as a real representation. When \(\pi \)  is orthogonal, there is a unique real representation \((\pi _0,V_0)\) , up to isomorphism, so that \(\pi \cong \pi _0 \otimes _{\mathbb {R}} . Observe that given a real representation \)\( of a finite group is automatically equivalent to an orthogonal representation. For example one can take the invariant non-degenerate symmetric bilinear form to be \)\(B(v,w)= \sum \limits _{g \in G} \langle g\cdot v, g\cdot w \rangle ,\)\(where \)\( is the standard inner product on \) Rn\(.\)
